Keywords: MHD system, sharp non-uniqueness, global solutions.

Mathematics Subject Classification: 35Q30, 35Q35, 76D03.

1 Introduction↩︎

In this paper, we consider the incompressible MHD equations on \(\mathbb{R}^2\) : \[\left\{ \begin{align} & \partial_t u-\Delta u+(u\cdot\nabla) u+\nabla p=B\cdot\nabla B,\\ & \partial_t B-\Delta B+u\cdot\nabla B =B\cdot\nabla u,\\ &\nabla\cdot u=\nabla \cdot B = 0, \end{align} \right. \label{MHD}\tag{1}\] supplemented with a divergence-free initial datum \((u^{\mathrm{in}},B^{\mathrm{in}})\). Here \(u\) is the velocity field, \(B\) the magnetic field and \(p\) the pressure.

For the Navier–Stokes equations (i.e., \(B=0\) in 1 ), Leray’s pioneering work [1] demonstrated the existence of global weak solutions for any initial data \(u^{\mathrm{in}}\in L^2(\mathbb{R}^3)\). These solutions obey the energy inequality \[\|u(t)\|_{L^2}^{2}+2\int_{0}^{t}\|\nabla u(s)\|_{L^2}^{2}\,\mathrm{d}s \leq \|u^{\mathrm{in}}\|_{L^2}^{2},\qquad \forall\, t\ge 0.\] Later, Hopf [2] extended this existence theory to smooth bounded domains in any dimension \(d\ge 2\) under Dirichlet boundary conditions. Such weak solutions are now commonly referred to as Leray–Hopf weak solutions. For dimension \(d\ge 3\), however, the question of uniqueness and regularity for Leray-Hopf weak solutions remains open.

The natural scaling of the Navier–Stokes equations in \(\mathbb{R}^d\) is given by \[u(t,x) \mapsto \lambda u(\lambda^2 t, \lambda x),\qquad p(t,x) \mapsto \lambda^2 p(\lambda^2 t, \lambda x),\quad \lambda>0.\] Under this scaling, local well-posedness results for the Cauchy problem have been established in various scaling-invariant spaces (see, e.g., [3]–[9]). The critical scaling in the mixed Lebesgue spaces \(L^q([0,T]; L^p(\mathbb{R}^d))\) leads to the Ladyzhenskaya–Prodi–Serrin (LPS for short) condition \[\begin{align} \label{LPS} \frac{2}{q}+\frac{d}{p}\le 1,\quad d\le p\le\infty. \end{align}\tag{2}\] If a Leray–Hopf weak solution \(u\) belongs to \(L^q_tL^p_x\) with 2 , then the solution is unique for all \(p\ge d\) and smooth for \(p>d\) [10]–[13]. For dimension \(d=3\), smoothness also holds at the endpoint \((q,p)=(\infty, 3)\), as proved by Escauriaza–Seregin–Šverák [14]. In fact, for very weak solutions, the LPS condition 2 excluding the endpoint \(p=d=3\) also provides a uniqueness criterion [15], [16], while the endpoint case \(p=d=3\) requires the stronger space \(C([0,T];L^3(\mathbb{R}^3))\) [17].

On the non-uniqueness of weak solutions to the Navier–Stokes equations, one elegant strategy developed by Jia and Šverák [18], [19] exploits the nonlinear instability of a steady solution in the self-similar dynamics. In the forced case, Albritton, Brué, and Colombo [20] adapted Vishik’s construction [21], [22] to prove non-uniqueness of Leray–Hopf weak solutions, consistent with the Jia–Šverák framework. Recently, Hou, Wang, and Yang [23] claim to prove non-uniqueness of Leray–Hopf solutions for the unforced 3D Navier–Stokes equations via a computer-assisted verification of key spectral properties. Another major direction stems from convex integration scheme, which originates in Nash’s \(C^1\) isometric embedding theorem [24]. It was first introduced into the Euler equations in the seminal works of De Lellis and Székelyhidi [25], [26], and subsequently extended to study the non-uniqueness of solutions for the Navier–Stokes equations. For example, on the periodic setting \(\mathbb{T}^d\), Buckmaster and Vicol [27] constructed the first non-unique weak solutions to the Navier–Stokes equations in \(C_t L^2\) for \(d=3\). Cheskidov–Luo proved non-uniqueness in \(L^\infty_t L^p\) (\(1\le p<2\)) for \(d=2\) [28], and in \(L^q_t L^\infty\) (\(1\le q<2\)) for \(d\ge 2\) [29], thereby demonstrating the sharpness of the LPS criteria in the endpoint cases \((q,p)=(\infty,2)\) and \((2,\infty)\), respectively. In contrast, on the whole space \(\mathbb{R}^3\), the authors in [30], [31] develop a new iterative framework to prove non-uniqueness in the classes \(C_t L^2\) and \(L^q_t L^\infty\) for \(1\le q<2\). In a separate development, Coiculescu and Palasek [32] recently pioneered a novel mechanism to construct two distinct global solutions in \({\rm BMO}^{-1}(\mathbb{T}^3)\).

For the MHD equations 1 , the existence of weak solutions in the Leray–Hopf class were established  [33], [34], while the questions of uniqueness and regularity remain open. Analogous to the Navier–Stokes equations, one may expect that if a Leray–Hopf calss of weak solution \((u,B)\) to 1 in \(L^q_tL^p_x\) with LPS condition 2 is unique. For the weak solutions, the LPS condition (excluding the endpoint \(p=d=3\)) also serves as a uniqueness criterion. At the endpoint \(q=d=3\), the general weak solution \((u,B)\) is unique in the smaller space \(C_t L^3\). The definition of the weak solutions is as follows:

Since the LPS condition 2 serves as a uniqueness criterion for weak solutions of the MHD equations 1 , a natural question arises: (Q): Is the LPS criterion 2 sharp for the uniqueness of weak solutions \((u,B)\) of 1 with a non‑trivial magnetic field? Alternatively, are such weak solutions \((u,B)\) of 1 non-unique in the class \(L^q_t L^p_x\) with \(\frac{2}{q} + \frac{d}{p} > 1\)?

The non-uniqueness problem for the MHD equations has also drawn considerable attention in the literature. For the ideal MHD system, smooth solutions conserve three important quantities: the total energy \(\mathcal{E}\), the cross helicity \(\mathcal{H}_c\), and the magnetic helicity \(\mathcal{H}_m\). This intrinsic feature naturally directs attention to the critical regularity thresholds for the conservation of these invariants. The first construction of nontrivial weak solutions violating total energy conservation was given by Bronzi, Lopes Filho, and Nussenzveig Lopes [35]. Subsequently, Faraco, Lindberg, and Székelyhidi [36] constructed compactly supported \(L^\infty\) weak solutions in \(\mathbb{R}^3\) for which \(\mathcal{E}\) and \(\mathcal{H}_c\) (the Elsässer energies) are not conserved, while \(\mathcal{H}_m\) vanishes identically. The first non‑conservative weak solutions with non‑trivial \(\mathcal{H}_m\), obtained by Beekie, Buckmaster, and Vicol [37], lie in \(C_t L^p(\mathbb{T}^3)\) for some \(p>2\) and fail to conserve \(\mathcal{E}\), \(\mathcal{H}_c\), and \(\mathcal{H}_m\). Later, Faraco, Lindberg, and Székelyhidi [38] refined their earlier construction and identified \(L^3_{t,x}\) as the sharp threshold for magnetic helicity conservation. The present authors in [39] constructed \(C^{1/3-}_{t,x}\) weak solutions on \(\mathbb{T}^3\) and \(C^{1/5-}_{t,x}\) weak solutions on \(\mathbb{T}^2\), both of which do not conserve the Elsässer energies, thereby resolving the flexible part of the Onsager-type conjecture for Elsässer energies in the three-dimensional case. More recently, Enciso, Peñafiel-Tomás, and Peralta-Salas [40] established the first Hölder continuous solutions that maintain a non-zero \(\mathcal{H}_m\) without conserving the Elsässer energies.

For the viscous and resistive MHD equations, the existence of various weak solutions is well understood, yet uniqueness is far from settled. In particular, for Leray–Hopf weak solutions with \(d \ge 3\), and for the weak solutions defined in [def-weak] with \(d \ge 2\), the uniqueness problem remains open. Only a few works have been devoted to the study of non‑uniqueness of weak solutions for the MHD system. Li, Zeng, and Zhang [41] proved the non-uniqueness of weak solutions in \(H^\varepsilon_{t,x}\) for some small \(\varepsilon\) by constructing new velocity and magnetic flows with refined spatial-temporal intermittency that respect the MHD geometry and control strong viscosity and resistivity. Subsequently, Miao and Ye [42] introduced box flows and constructed perturbations consisting of seven different types of flows to overcome dissipation, thereby proving the non-uniqueness of weak solutions in \(C_tL^2(\mathbb{T}^3)\) with controllable total energy and cross helicity. The authors of [43], [44] established the non-uniqueness of weak solutions in \(L^q L^\infty(\mathbb{T}^3)\) for \(1\le q<2\), which demonstrates the sharpness of the endpoint \((q,p)=(2,\infty)\) in the LPS criterion 2 for \(\mathbb{T}^d\) with \(d\ge 3\). However, the convex integration methods in [43], [44] break down in the two-dimensional setting. Recently, Dai [45] constructed solutions whose \(L^\infty\) norm blows up instantaneously, and showed non-uniqueness of spatial smooth solutions.

For weak solutions of the MHD equations 1 , the Serrin criterion is a sufficient condition for uniqueness when \(d\ge 2\). However, no results are available on the sharpness of this criterion in the two-dimensional case on the whole space. In this paper, we construct two distinct weak solutions sharing the same initial data to demonstrate the sharpness of the classical LPS uniqueness criterion for the MHD equations on \(\mathbb{R}^2\) at the endpoint \((2,\infty)\). Our result is as follows.

Below we compare different non-uniqueness results of the MHD equations 1 using the scales of space-time Lebesgue spaces \(L^q_tL^p_x\).

As shown in the table, all previous non‑uniqueness results for the MHD equations 1 were obtained in the three‑dimensional setting via convex integration methods. This is primarily because the higher dimension offers more choices of oscillation directions for constructing building blocks, thereby rendering the convex integration scheme more feasible in three dimensions than in two. For the two‑dimensional ideal MHD equations, as shown in [36], [47], the \(\Lambda\)-convex hull \(K^{\Lambda}\) has an empty interior, which consequently leaves “little room” for a convex integration argument.

To our knowledge, the only convex integration result in two dimensions for the MHD equations to date is the construction by [39] of Hölder weak solutions that exhibit energy dissipation in the ideal (i.e., inviscid and non‑resistive) case. The construction in [39] exploits the finite propagation speed property. Nevertheless, it is not applicable to the viscous and resistive MHD system. We develop a new mechanism that is entirely different from those used in previous non‑uniqueness results for the MHD equations. In what follows, we describe our main ideas in more details. The construction in Theorem 1 is based on a new iterative scheme that yields exact solutions to the MHD equations 1 at each iteration step. This scheme differs fundamentally from the approaches used in previous works on non‑uniqueness for the MHD equations [41]–[44], where one constructs approximate solutions to a relaxed system rather than exact solutions to the original equations. More precisely, we give a solution \((u_1, B_1)\) of system 1 , and assume that for \(n \ge 3\) we already have \(\{u_j, B_j\}_{1 \le j < n}\) as solutions to the MHD equations 1 . Our main task is to construct \((u_n, B_n)\) by building an increment \((w_n, d_n)\) on \((u_{n-2}, B_{n-2})\), so that the resulting pair \((u_n, B_n)\) remains a solution to the MHD equations. The construction of \((w_n, d_n)\) consists of two parts: the first part is an explicit construction based on the previously known increment \(w_{n-1}\), and the second part is obtained via the solvability of a forced MHD equations.

The explicit construction consists of two components: the heat-dominated Fourier mode flow \({w}^{\text{(h)}} _n\) and the inverse cascade-dominated flow \({w}^{\text{(i)}} _n\). Since our target space is \(L^2_t L^p\) with \(p<\infty\), the intermittent building blocks employed in [43], [44] become ineffective when \(p>2\). Hence, the construction presented here is completely different from those in previous works. The main part \({w}^{\text{(h)}} _n\) is constructed as follows: \[{w}^{\text{(h)}} _n \sim \lambda_n e^{-\lambda_n^2 t} \sum_{k\in\Lambda} a_k \left( \mathrm{Id} + \frac{\mathcal{R} w_{n-1}}{1000\|\mathcal{R} w_{n-1}\|_{L^\infty}} \right) \eta_{n-1} \chi_{n-1} \, \phi(\lambda_n^r k\cdot x)\, e^{\mathrm{i}\lambda_n k\cdot x} \,\bar{k} \;+\; \text{l.o.t.},\] where \(\phi(\lambda_n^r k\cdot x)\bar{k}\) is a two-dimensional Mikado flow and the exponent \(r\) is dependent on \(p\). Inspired by  [32], we introduce the Mikado flow together with the smooth cutoff function \(\chi\) so that the support of \({w}^{\text{(h)}} _n\) shrinks monotonically as the iteration proceeds, which enables us to achieve a limiting solution belonging to \(L^2_t L^p\) (\(p<\infty\)). The 2D Mikado flows inevitably intersect geometrically, thereby producing a delicate error from the quadratic form \(\text{div}( {w}^{\text{(h)}} _n\otimes {w}^{\text{(h)}} _n)\). To overcome this, we introduce a highly oscillatory flow \(e^{-\lambda_n^2 t}\sum_{k\in\Lambda} e^{\mathrm{i}\lambda_n k\cdot x}\bar{k}\) into \({w}^{\text{(h)}} _n\). This flow is designed to serve two crucial purposes simultaneously:

• Absorption into pressure – the highly oscillatory flow is an exact solution of the steady two-dimensional Euler equations, so that the oscillatory part of \(\text{div}( {w}^{\text{(h)}} _n\otimes {w}^{\text{(h)}} _n)\) can be absorbed into the pressure term while keeping the residual error under control.

• Cancellation of the principal dissipative error – the highly oscillatory flow is also a solution of the free heat equation, thereby eliminating the main error caused by the dissipative term, in contrast to previous convex integration iterations where \(-\Delta u\) is handled as an error term.

The inverse cascade-dominated flow \({w}^{\text{(i)}} _n\) is given by \({w}^{\text{(i)}} _{n}= {w}^{\text{(h)}} _{n-1}e^{-2\lambda^2_{n}t}\), as introduced by [32]. The construction of \({w}^{\text{(i)}} _{n}\) guarantees that the \(\lambda_n\)-level oscillatory error generated by the Mikado flows in the quadratic term \(\text{div}( {w}^{\text{(h)}} _n\otimes {w}^{\text{(h)}} _n)\) is cancelled by \(\partial_t {w}^{\text{(i)}} _n\), leaving only lower-order parts.

The implicit part of the increment \((w_n, d_n)\) is the perturbation flow \(( {w}^{\text{(m)}} _n, {d}^{\text{(m)}} _n)\). Our key insight is to fully exploit the exponential decay of the time factor \(e^{-\lambda_n^2 t}\) in the construction of \({w}^{\text{(h)}} _n\) and \({w}^{\text{(i)}} _n\). This allows us to expect that the \(L^1_t C^1\) norm of \((u_{n-2},B_{n-2})\) grows only linearly in \(n\) (i.e., \(O(n)\)). At the same time, the residual error decays doubly exponentially (e.g., \(\lambda_{n-1}^{-30}\)) in the \(L^1_t B^{-1+\frac{2}{p}}_{p,1}\) space. The latter decays far faster than the former grows, which allows us to treat the residual as a forcing term. Consequently, by solving a forced MHD system with a suitably chosen small initial datum (see 38 ), we obtain a small corrector \(( {w}^{\text{(m)}} _n, {d}^{\text{(m)}} _n)\). Finally, let \((w_n, d_n) = ( {w}^{\text{(h)}} _n + {w}^{\text{(i)}} _n + {w}^{\text{(m)}} _n, {d}^{\text{(m)}} _n)\) and we can construct \((u_n, B_n)\) to be an exact solution of the MHD equations.

Organization of the paper. Section 2 collects the necessary preliminaries and technical tools that will be used throughout the paper. In Section 3, we present an iterative framework and state the key iteration proposition (Proposition 4); we then show that Theorem 1 follows directly from this proposition. Finally, the proof of Proposition 4 is given in Section 4.

Notation. In this paper, \(\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2\). For a \(\mathbb{T}^2\)-periodic function \(f\), we denote \[\begin{align} \mathbb{P}_{=0} f:=\frac{1}{|\mathbb{T}^2|}\int_{\mathbb{T}^2} f(x)\mathop{}\!\mathrm{d}x,\quad\mathbb{P}_{\neq 0}f:=f-\mathbb{P}_{=0} f. \end{align}\] In the following, the notation \(a\lesssim b\) means \(a\le Cb\) for a universal constant \(C\) that may change from line to line. Without ambiguity, we will denote \(L^r([0,T];Y(\mathbb{T}^2))\) and \(L^r([0,\infty);Y(\mathbb{T}^2))\) by \(L^r_T Y\) and \(L^r_t Y\), respectively. For a vector funtion \(f=(f_1, f_2)\), \(\mathop{\mathrm{curl}}f=\partial_1 f_2-\partial_2 f_1\).

Let \(\partial^{\sigma}\) be the space derivatives, we denote \[\begin{align} \|f\|_{L^r_tC^N}:=\sum_{j=0}^N\max_{|\sigma|=j}\|\partial^\sigma f\|_{L^r_{t}L^\infty}. \end{align}\] and \[\begin{align} \|(f_1,f_2,\cdots,f_m)\|_{X}:=\max\Big\{\|f_1\|_X, \|f_2\|_X, \cdots, \|f_m\|_X\Big\}. \end{align}\]

2 Preliminaries↩︎

In this section we collect several useful tools: the definitions of mollifiers, a geometric lemma, Besov spaces and Chemin–Lerner spaces, as well as some estimates for transport‑diffusion equations and oscillating exponential factors.

We now recall the definitions of Besov spaces and the mixed time‑space Besov spaces (the so‑called Chemin–Lerner spaces).

3 Induction scheme↩︎

Without loss of generality, we first aim to establish Theorem 1 for a fixed exponent \(2<p<\infty\). To this end, we construct a new iterative scheme.

3.1 Parameters↩︎

First of all, we introduce several parameters. Given \(f\in C^\infty_0(\mathbb{R}^2)\), \(2<p<\infty\) and \(T>0\), we first choose \(r\in\mathbb{Q}\) such that \[0 < r < \frac{1}{10}\Bigl(1-\frac{2}{p}\Bigr).\] Then there exists a positive integer \(M\) with \(rM\in\mathbb{N}\). Since the set \(\Lambda\subseteq \mathbb{S}^1\cap\mathbb{Q}^2\) in Lemma 2 is finite, we can find a positive integer \(N_{\Lambda}\) such that \[N_{\Lambda} k \in \mathbb{Z}^2,\qquad \forall k\in\Lambda.\] Now we introduce two large positive integers \(a,b\in M\mathbb{N}\) that will be chosen later depending on \(T\), \(C_0\), \(p\), and \(\|f\|_{\dot{B}^{0}_{2,1}\cap \dot{B}^{1}_{2,1}\cap B^{-1}_{1,1}}\) and satisfy \[b > 2^{10}\max\Bigl\{\bigl(1-\tfrac{2}{p}\bigr)^{-1},\; \tfrac{p}{2}\Bigr\}, \qquad a > \|f\|_{\dot{B}^{0}_{2,1}\cap \dot{B}^{1}_{2,1}\cap B^{-1}_{1,1}}.\] For any \(q\ge 0\), we define \[\begin{align} \label{def-ell} \lambda_q := N_{\Lambda}^{M} a^{b^q}, \qquad \ell_q := \min\Bigl\{\lambda_q^{-\frac{50}{1-\frac{2}{p}}},\; \lambda_q^{-\frac{50}{\frac{2}{p}}}\Bigr\}. \end{align}\tag{3}\]

3.2 Iterative procedure↩︎

We now construct the iteration scheme by induction. For each integer \(j\ge 0\), set \[\mathcal{O}_j = \Bigl[-\tfrac{3}{4}\pi + 5\lambda_j^{-\frac{1}{50}},\; \tfrac{3}{4}\pi - 5\lambda_j^{-\frac{1}{50}}\Bigr]^2,\] and let \(\eta_j \in C_c^\infty(\mathbb{R}^2)\) be a smooth cut-off function such that \[\begin{align} \label{def-eta} \text{spt}\,\eta_j\subseteq\mathcal{O}_{j+1} , \qquad \eta_j(x) \equiv 1 \;\; \text{for all } x \in \mathcal{O}_{j}. \end{align}\tag{4}\] Define \(\varepsilon_0 = \lambda_1^{-\frac{1}{15}}\) and denote the orthogonal gradient by \(\nabla^{\perp} = (\partial_2, -\partial_1)\). We introduce \(( {w}^{\text{(h)}} _1, {w}^{\text{(i)}} _1)\) as \[{w}^{\text{(h)}} _1 = \varepsilon_0 \lambda_1^{-3} e^{-\lambda_1^2 t} \, \Delta \nabla^{\perp} \Bigl( \eta_0 \operatorname{curl}\bigl( \sin(\lambda_1 x_1) e_2 \bigr) \Bigr), \qquad {w}^{\text{(i)}} _1 = 0.\]

Let \(( {w}^{\text{(m)}} _1, p^{(\mathrm{m})}_1, {d}^{\text{(m)}} _1)\) be the unique global solution of the forced MHD system \[\label{def-w1} \left\{ \begin{align} &\partial_t {w}^{\text{(m)}} _1 - \Delta {w}^{\text{(m)}} _1 + {w}^{\text{(m)}} _1\!\cdot\!\nabla {w}^{\text{(m)}} _1 + {w}^{\text{(h)}} _1\!\cdot\!\nabla {w}^{\text{(m)}} _1 + {w}^{\text{(m)}} _1\!\cdot\!\nabla {w}^{\text{(h)}} _1 + \nabla p^{(\mathrm{m})}_1 \\ &= {d}^{\text{(m)}} _1\!\cdot\!\nabla {d}^{\text{(m)}} _1 - (\partial_t - \Delta) {w}^{\text{(h)}} _1 - \text{div}( {w}^{\text{(h)}} _1 \otimes {w}^{\text{(h)}} _1), \\[4pt] &\partial_t {d}^{\text{(m)}} _1 - \Delta {d}^{\text{(m)}} _1 + ( {w}^{\text{(m)}} _1 + {w}^{\text{(h)}} _1)\!\cdot\!\nabla {d}^{\text{(m)}} _1 = {d}^{\text{(m)}} _1\!\cdot\!\nabla ( {w}^{\text{(m)}} _1 + {w}^{\text{(h)}} _1), \\[4pt] &\nabla\!\cdot\! {w}^{\text{(m)}} _1 = \nabla\!\cdot\! {d}^{\text{(m)}} _1 = 0, \\[4pt] &( {w}^{\text{(m)}} _1, {d}^{\text{(m)}} _1)\big|_{t=0} = \bigl(0,\; \lambda_1^{-50} f(x) \bigr). \end{align} \right.\tag{5}\] Owing to the specific choice of \({w}^{\text{(h)}} _1\) and the smallness of \(\varepsilon_0\), the system 5 admits a unique global solution \(( {w}^{\text{(m)}} _1, p^{(\mathrm{m})}_1, {d}^{\text{(m)}} _1)\) satisfying the estimates \[\begin{align} \|( {w}^{\text{(m)}} _1, {d}^{\text{(m)}} _1)\|_{\widetilde{L}^{\infty}_t (\dot{B}^{-1+\frac{2}{p}}_{p,1} \cap \dot{B}^{-\frac{2}{p}}_{2,1})\cap L^{1}_t (\dot{B}^{1+\frac{2}{p}}_{p,1} \cap \dot{B}^{2-\frac{2}{p}}_{2,1})} \lesssim{\varepsilon_0 }(\lambda^{-1+\frac{2}{p}}_1+\lambda^{-\frac{2}{p}}_1), \label{wmhd1} \end{align}\tag{6}\] and \[\begin{align} \|( {w}^{\text{(m)}} _1, {d}^{\text{(m)}} _1)\|_{L^{\infty}_T B^{-1}_{1,1} \cap L^{1}_T {B}^{1}_{1,1}} \lesssim\varepsilon_0,\qquad \|( {w}^{\text{(m)}} _1, {d}^{\text{(m)}} _1)\|_{\widetilde{L}^{\infty}_t \dot{B}^{0}_{\infty,1} \cap L^{1}_t \dot{B}^{2}_{\infty,1}} \leq \lambda_1^{-\frac{1}{20}}. \label{wmhd1-1} \end{align}\tag{7}\] Finally we define \((u_1, p_1, B_1)\) by \[(u_1, p_1, B_1) = \bigl( {w}^{\text{(h)}} _1 + {w}^{\text{(m)}} _1,\; p^{(\mathrm{m})}_1,\; {d}^{\text{(m)}} _1 \bigr)\] that satisfies the MHD equations 1 .

Now assume that for some integer \(n\ge 2\) and each \(1\le j\le n-1\), we have already constructed smooth solutions \(\{(u_j,B_j)\}_{1\le j\le n-1}\subseteq C^\infty_{t,x}(\mathbb{R}^{+}\times \mathbb{R}^2)\) of 1 on \(\mathbb{R}^{+}\times\mathbb{R}^2\) satisfying \[\begin{align} & \|(u_j, B_j)\|_{L^2_TL^2}\le 2C^{3/4}_0\sum_{k=1}^j 2^{-\frac{j}{2}},\quad \|(u_j, B_j)\|_{L^\infty_t\dot{B}^0_{\infty,1}\cap L^1_t\dot{B}^2_{\infty,1}}\le C_0\lambda_j,\tag{8} \\[4pt] & \|(u_j,B_j)\|_{L^1_t\dot{B}^{1}_{\infty,1}\cap L^2_t \dot{B}^0_{\infty,1}} \le \begin{cases} C_0 q, & \text{if } j = 2q, \\[2pt] C_0 (q+1), & \text{if } j = 2q+1, \end{cases} \tag{9} \\[8pt] & u_j = \begin{cases} \displaystyle \sum_{l=1}^{q} w_{2l} = \sum_{l=1}^{q} \bigl( {w}^{\text{(h)}} _{2l} + {w}^{\text{(i)}} _{2l} + {w}^{\text{(m)}} _{2l} \bigr), & \text{if } j = 2q, \\[10pt] \displaystyle \sum_{l=0}^{q} w_{2l+1} = \sum_{l=0}^{q} \bigl( {w}^{\text{(h)}} _{2l+1} + {w}^{\text{(i)}} _{2l+1} + {w}^{\text{(m)}} _{2l+1} \bigr), & \text{if } j = 2q+1, \end{cases} \tag{10} \end{align}\] and \[\begin{align} & B_j = \begin{cases} \displaystyle \sum_{l=1}^{q} {d}^{\text{(m)}} _{2l}, & \text{if } j = 2q, \\[10pt] \displaystyle \sum_{l=0}^{q} {d}^{\text{(m)}} _{2l+1} , & \text{if } j = 2q+1. \end{cases} \label{b-decomposition} \end{align}\tag{11}\] Set \(\mathcal{R} := (\nabla+\nabla^{\!\top})(-\Delta)^{-1}\). Then \({w}^{\text{(h)}} _j\) and \({w}^{\text{(i)}} _j\) can be rewritten as \[\begin{align} \label{def-R} {w}^{\text{(h)}} _j = \text{div}(\mathcal{R} {w}^{\text{(h)}} _j),\qquad {w}^{\text{(i)}} _j = \text{div}(\mathcal{R} {w}^{\text{(i)}} _j) \end{align}\tag{12}\] and satisfy \[\begin{align} &\|\partial_t^{\,l}\, \mathcal{R} {w}^{\text{(h)}} _j\|_{L^\infty _tC^{N}} \le C^{3/4}_0 \lambda_j^{N+2l}, \qquad \|\mathcal{R} {w}^{\text{(i)}} _j(t)\|_{L^\infty_tC^{N}} \le C^{3/4}_0 \lambda_{j-1}^{N},\qquad 0\le N\le 3,\; l=0,1,\tag{13}\\ & \| {w}^{\text{(h)}} _j\|_{L^{2}_tL^p \cap L^{1}_t W^{1,p}} \le C_0^{3/4}\, 2^{-\frac{j}{p}}, \qquad \| {w}^{\text{(i)}} _j\|_{L^{2}_tL^q \cap L^{1}_t W^{1,q}} \le C_0^{3/4} 2^{-\frac{j}{p}},\qquad \forall\, 1\le q\le \infty, \tag{14} \end{align}\] and for each \(t\ge 0\), \[\begin{align} &\|w^{(\rm h)}_{j}(t)\|_{ \dot{B}^{N}_{\infty,1}} \leq C^{3/4}_0 e^{-\lambda^2_{j}t}\lambda^{N+1}_{j}, \quad \| {w}^{\text{(i)}} _j(t)\|_{ \dot{B}^{N}_{\infty,1}} \leq C^{3/4}_0 e^{-(\lambda^2_j+\lambda^2_{j-1}) t} \lambda^{N+1}_{j-1},\quad -2\le N\le 2. \label{wp-wsL1} \end{align}\tag{15}\] Moreover, the perturbation flows \(( {w}^{\text{(m)}} _j, {d}^{\text{(m)}} _j)\) obey \[\begin{align} &\|( {w}^{\text{(m)}} _j, {d}^{\text{(m)}} _j)\|_{\widetilde{L}^{\infty}_t\dot{B}^{ -1+\frac{2}{p}}_{p,1}\cap L^{1}_t\dot{B}^{1+\frac{2}{p}}_{p,1}} \le \lambda^{-20}_{j-1},\quad \|( {w}^{\text{(m)}} _j, {d}^{\text{(m)}} _j)\|_{\widetilde{L}^{\infty}_t\dot{B}^{ 0}_{\infty,1}\cap L^{1}_t\dot{B}^{2}_{\infty,1}} \le \lambda^{2r}_{j}. \label{e-wmdm} \end{align}\tag{16}\]

We now show that Theorem 1 follows directly from Proposition 4.

Proposition 4 implies Theorem 1. Applying Proposition 4 repeatedly generates two families of smooth solutions \(\{(u_{2q-1},B_{2q-1})\}_{q\ge 1}\) and \(\{(u_{2q},B_{2q})\}_{q\ge 1}\) to the MHD system 1 , given explicitly by \[\begin{align} (u_{2q-1}, B_{2q-1}) = \Bigl( \sum_{l=1}^{q} w_{2l-1},\; \sum_{l=1}^{q} d^{\rm (m)}_{2l-1} \Bigr), \,\,\text{and}\,\, (u_{2q}, B_{2q}) = \Bigl( \sum_{l=1}^{q} w_{2l},\; \sum_{l=1}^{q} d^{\rm (m)}_{2l} \Bigr). \end{align}\]

Since \(p>2\), we have the continuous embeddings \[\begin{align} \widetilde{L}^{\infty}_t\dot{B}^{-\frac{2}{p}}_{2,1}(\mathbb{R}^2)\cap L^{1}_t\dot{B}^{2-\frac{2}{p}}_{2,1}(\mathbb{R}^2) &\hookrightarrow \widetilde{L}^{\infty}_t\dot{B}^{-1}_{p,1}(\mathbb{R}^2)\cap L^{1}_t\dot{B}^{1}_{p,1}(\mathbb{R}^2) \hookrightarrow L^{2}_t L^p(\mathbb{R}^2) \cap L^{1}_t W^{1,p}(\mathbb{R}^2), \end{align}\] and \[L^\infty_T B^{-1}_{1,1}(\mathbb{R}^2)\cap L^1_T B^1_{1,1}(\mathbb{R}^2) \hookrightarrow L^2_T B^0_{1,1}(\mathbb{R}^2)\cap L^1_T B^1_{1,1}(\mathbb{R}^2) \hookrightarrow L^2_T L^1(\mathbb{R}^2)\cap L^1_T W^{1,1}(\mathbb{R}^2).\] Combining these embeddings with the estimates 14 and ?? yields \[\begin{align} \sum_{q=1}^\infty \bigl\| (w_{2q}, w_{2q-1}, d^{(\rm m)}_{2q}, d^{(\rm m)}_{2q-1}) \bigr\|_{L^2_t L^p \cap L^1_t W^{1,p} \cap L^2_T L^1 \cap L^1_T W^{1,1}} \le \sum_{q=1}^\infty \bigl( 2C_0^{3/4} 2^{-\frac{2q}{p}} + \lambda_{2q-1}^{-10} \bigr) < \infty. \end{align}\] In view of 6 , 7 and ?? , the above estimate implies that both \(\{(u_{2q-1},B_{2q-1})\}_{q\ge 1}\) and \(\{(u_{2q},B_{2q})\}_{q\ge 1}\) are Cauchy sequences in the space \[L^{2}_t L^p(\mathbb{R}^2) \cap L^{1}_t W^{1,p}(\mathbb{R}^2) \cap L^2_T L^1(\mathbb{R}^2) \cap L^1_T W^{1,1}(\mathbb{R}^2).\] Hence there exist the two limit functions \[(u, B),\;(\widetilde{u}, \widetilde{B}) \in L^2\bigl([0,T]; L^1(\mathbb{R}^2)\cap L^p(\mathbb{R}^2)\bigr) \cap L^1\bigl([0,T]; W^{1,1}(\mathbb{R}^2)\cap W^{1,p}(\mathbb{R}^2)\bigr)\] such that, as \(q\to\infty\), \[(u_{2q-1},B_{2q-1}) \to (u, B), \qquad (u_{2q}, B_{2q}) \to (\widetilde{u}, \widetilde{B}),\] strongly in \(L^{2}_T(L^1(\mathbb{R}^2)\cap L^p(\mathbb{R}^2)) \cap L^{1}_T(W^{1,1}(\mathbb{R}^2)\cap W^{1,p}(\mathbb{R}^2))\). Moreover, both \((u, B)\) and \((\widetilde{u}, \widetilde{B})\) satisfy the MHD system 1 .

From the initial conditions ?? , we deduce \[u_{2q}(0,x) = u_{2q+1}(0,x) - {w}^{\text{(h)}} _{2q+1}(0,x), \qquad B_{2q}(0,x) = B_{2q+1}(0,x) = \sum_{j=1}^q \lambda_{2j-1}^{-50} f(x).\] Thanks to 15 , the term \({w}^{\text{(h)}} _{2q+1}(0,x)\to 0\) in \(\dot{B}^{-2}_{\infty,1}(\mathbb{R}^2)\) as \(q\to\infty\). Consequently, \[\bigl(u(0,x), B(0,x)\bigr) = \bigl(\widetilde{u}(0,x), \widetilde{B}(0,x)\bigr) \qquad \text{in the sense of distributions}.\]

It remains to prove that \((u, B) \neq (\widetilde{u}, \widetilde{B})\). Because both solutions belong to \(C^\infty(\mathbb{R}^{+}\times \mathbb{R}^2)\), it suffices to show that \[\|(u, B)(\lambda_1^{-2}, \cdot)\|_{\dot{B}^{-1}_{\infty,1}} \neq \|(\widetilde{u}, \widetilde{B})(\lambda_1^{-2}, \cdot)\|_{\dot{B}^{-1}_{\infty,1}}.\]

Recall that \[u_1 = {w}^{\text{(h)}} _{1} + {w}^{\text{(m)}} _1 = \varepsilon_0 \lambda_1^{-3} e^{-\lambda_1^2 t} \Delta \nabla^{\perp} \bigl( \eta_0 \operatorname{curl}(\sin(\lambda_1 x_1) e_2) \bigr) + {w}^{\text{(m)}} _1.\] At \(t = \lambda_1^{-2}\), we obtain the lower bound \[\begin{align} \bigl\| {w}^{\text{(h)}} _{1}(\lambda_1^{-2},\cdot)\bigr\|_{\dot{B}^{-1}_{\infty,1}} &\ge C\varepsilon_0 e^{-1} \lambda_1^{-3} \bigl\| \Delta \bigl( \eta_0 \operatorname{curl}(\sin(\lambda_1 x_1) e_2) \bigr) \bigr\|_{\dot{B}^{0}_{\infty,1}} \\ &\ge C\varepsilon_0 e^{-1} \lambda_1^{-3} \bigl\| \eta_0 \Delta \operatorname{curl}(\sin(\lambda_1 x_1) e_2) \bigr\|_{\dot{B}^{0}_{\infty,1}} \\ &\quad - C\varepsilon_0 e^{-1} \lambda_1^{-3} \Bigl( \bigl\| (\Delta\eta_0) \operatorname{curl}(\sin(\lambda_1 x_1) e_2) \bigr\|_{\dot{B}^{0}_{\infty,1}} \\ &\qquad\quad + 2\sum_{l=1}^2 \bigl\| \partial_l\eta_0 \, \partial_l \operatorname{curl}(\sin(\lambda_1 x_1) e_2) \bigr\|_{\dot{B}^{0}_{\infty,1}} \Bigr) \\ &\ge \frac{C}{2} \varepsilon_0 e^{-1} =: M_0. \end{align}\] Using 6 , we obtain \[\begin{align} \bigl\| u_{1}(\lambda_1^{-2},\cdot) \bigr\|_{\dot{B}^{-1}_{\infty,1}} &\ge \bigl\| {w}^{\text{(h)}} _{1}(\lambda_1^{-2},\cdot) \bigr\|_{\dot{B}^{-1}_{\infty,1}} - \bigl\| {w}^{\text{(m)}} _1 \bigr\|_{\dot{B}^{-1}_{\infty,1}} \ge \frac{C}{2} \varepsilon_0 e^{-1} - C\varepsilon_0 \bigl( \lambda_1^{-1+\frac{2}{p}} + \lambda_1^{-\frac{2}{p}} \bigr) \ge \frac{1}{2} M_0, \end{align}\] provided that the integer \(a\) is chosen sufficiently large. Thanks to 15 and 16 , we have \[\begin{align} \sum_{q\ge 1} \bigl\| ( {w}^{\text{(h)}} _{2q}, {w}^{\text{(i)}} _{2q}, {w}^{\text{(m)}} _{2q}, {w}^{\text{(h)}} _{2q+1}, {w}^{\text{(i)}} _{2q+1}, {w}^{\text{(m)}} _{2q+1})(\lambda_1^{-2},\cdot) \bigr\|_{\dot{B}^{-1}_{\infty,1}} \lesssim \sum_{q\ge 1} \bigl( C_0^{2} e^{-\lambda_{2q}^2 \lambda_1^{-2}} + \lambda_{2q-1}^{-10} \bigr) \lesssim \lambda_1^{-8}, \end{align}\] where the last inequality holds for large enough \(a\). Consequently, \[\begin{align} &\bigl\| (u- \widetilde{u})(\lambda_1^{-2},\cdot) \bigr\|_{\dot{B}^{-1}_{\infty,1}}\\ \ge& \bigl\| {w}^{\text{(h)}} _{1}(\lambda_1^{-2},\cdot) \bigr\|_{\dot{B}^{-1}_{\infty,1}} - \sum_{q\ge 1} 6 \bigl\| ( {w}^{\text{(h)}} _{2q}, {w}^{\text{(m)}} _{2q}, {w}^{\text{(i)}} _{2q}, {w}^{\text{(h)}} _{2q+1}, {w}^{\text{(i)}} _{2q+1}, {w}^{\text{(m)}} _{2q+1})(\lambda_1^{-2},\cdot) \bigr\|_{\dot{B}^{-1}_{\infty,1}} \\ \ge& \frac{1}{2} M_0 - C\lambda_1^{-8} \ge \frac{M_0}{4}, \end{align}\] as soon as \(a\) is sufficiently large. This completes the proof of Theorem 1. We now proceed to establish Proposition 4.

4 Proof of Proposition 4↩︎

Without loss of generality, we may assume that the index \(n\) is even; write \(n = 2q\) with \(q\in\mathbb{N}\). We shall construct the increment \((w_{2q}, {d}^{\text{(m)}} _{2q})\) to be added to the existing solution \((u_{2q-2}, B_{2q-2})\), thereby obtaining \((u_{2q}, B_{2q}) = (u_{2q-2} + w_{2q},\, B_{2q-2} + {d}^{\text{(m)}} _{2q})\). The velocity increment \(w_{2q}\) is decomposed into three components:

• the heat‑dominated Fourier mode flow \({w}^{\text{(h)}} _{2q}\),

• the inverse‑cascade dominated flow \({w}^{\text{(i)}} _{2q}\),

• the perturbation flow \({w}^{\text{(m)}} _{2q}\).

Thus \(w_{2q} = {w}^{\text{(h)}} _{2q} + {w}^{\text{(i)}} _{2q} + {w}^{\text{(m)}} _{2q}\), while the magnetic increment is simply \({d}^{\text{(m)}} _{2q}\).

4.1 Construction of the flows \({w}^{\text{(h)}} _{2q}\) and \({w}^{\text{(i)}} _{2q}\)↩︎

To ensure that \(\{ {w}^{\text{(h)}} _{2q}\}\) and \(\{ {w}^{\text{(i)}} _{2q}\}\) are Cauchy sequences in the integrable spaces \(L^p\), we introduce the cutoff function \(\chi_{2q-1}\). This idea, originating from [32], will be employed to obtain a monotonically decreasing sequence of compact support volumes in the iteration scheme. First, let \(\phi\) be a smooth, even function on the torus \(\mathbb{T}\) such that \(\int_{\mathbb{T}} \phi^2 \,\mathop{}\!\mathrm{d}x = 1\) and \(\operatorname{spt}\phi \subseteq \bigcup_{m \in \mathbb{Z}} \bigl([-\tfrac{1}{100}, \tfrac{1}{100}] + 2\pi m\bigr)\). The building blocks are then defined as \[\begin{align} \label{def-phi} \phi_{k,l}(x) = \phi\bigl( \lambda^{r}_{l} \, k \cdot x \bigr), \qquad l \in \mathbb{N}^{+}, \; k \in \Lambda, \end{align}\tag{17}\] where \(\Lambda\) is the finite set defined in Lemma 2. For \(m\ge 1\), we set \[\begin{align} \label{def-Omega} &\Omega_{m} = \bigcap_{1 \le l \le m} \, \bigcup_{k \in \Lambda} \operatorname{spt}\, \phi_{k,l}=:\bigcap_{1 \le l \le m} \, \bigcup_{k \in \Lambda} \Omega_{k,l},\notag\\ &\widetilde{\Omega}_{m}=\bigcap_{1 \le l \le m}\bigcup_{k \in \Lambda} \widetilde{\Omega}_{k,l},\qquad \widetilde{\Omega}_{k,l} := \bigl\{ x : d(x, \Omega_{k,l}) < \tfrac{1}{100} \lambda_l^{-r} \bigr\}. \end{align}\tag{18}\] Then a sequence \(\{\chi_m\}\) is defined recursively by: \(\chi_1 = 1\) on \(\Omega_1\) with \(\text{spt}\chi_1\subseteq\widetilde{\Omega_1}\). For \(m\ge 2\), \[\begin{align} \label{def-chi} \chi_m= 1 \text{ on } \text{spt}\,(\chi_{m-1}\phi_m) \quad\text{and}\quad \text{spt}\,\chi_m\subseteq\widetilde{\Omega}_m, \end{align}\tag{19}\] and these functions satisfy \(\|\chi_m\|_{C^N} \lesssim \lambda_m^{Nr}\) for \(m\ge 1, N\ge 0\).

We now define the heat‑dominated Fourier mode flow \({w}^{\text{(h)}} _{2q}\) by \[\label{def-wh} {w}^{\text{(h)}} _{2q} = -\lambda^{-3}_{2q} e^{-\lambda^2_{2q }t} \, \Delta \, \nabla^{\perp} \Bigl( \sum_{k \in \Lambda} a_{(k,2q)} \, \eta_{2q-1}\chi_{2q-1} \, \phi_{k,2q} \; \operatorname{curl} \bigl( e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}\bigr) \Bigr),\tag{20}\]

where

• \(\eta_{2q-1}\), \(\phi_{k,2q}\) and \(\chi_{2q-1}\) are defined in 4 , 17 and 19 , respectively;

• for each \(k\in\Lambda\), the amplitude \(a_{(k,2q)}\) is given by \[\begin{align} \label{def-akq} a_{(k,2q)}(t,x) &= (2000C_0)^{1/2} \int_{t}^{t+\ell_{2q-1}} \Bigl( a_k \bigl( \mathrm{Id} + \tfrac{ \mathcal{R}w^{(p)}_{2q-1} }{ 1000C_0} \bigr) \ast \psi_{\ell_{2q-1}} \Bigr)(x,s) \; \varphi_{\ell_{2q-1}}(t-s) \, \mathop{}\!\mathrm{d}s \end{align}\tag{21}\] with \(a_k\) the coefficients from Lemma 2 and \(\psi_{\ell}\), \(\varphi_{\ell}\) the mollifiers from Definition 2;

• from the definitions of \(a_{(k,2q)}\) and \(\phi_{k,2q}\) we immediately obtain, for any \(N\ge 0\), \[\begin{align} \label{est-aphi} \|a_{(k,2q)}\|_{L^\infty_t C^N_x} \lesssim (2000C_0)^{1/2} \, \ell_{2q-1}^{-N}, \qquad \|\phi_{k,2q}\|_{C^N} \lesssim \lambda^{rN}_{2q}; \end{align}\tag{22}\]

• since \(k \perp \bar{k}\) and \(|k|=1\), one has \(-\Delta \nabla^{\perp} \operatorname{curl} \bigl( e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}\bigr) = \lambda^4_{2q} e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}\), which yields the decomposition \[\begin{align} \label{dec-wh} {w}^{\text{(h)}} _{2q}= {w}^{\text{(h,m)}} _{2q} + {w}^{\text{(h,r)}} _{2q} \end{align}\tag{23}\] with \[\begin{align} {w}^{\text{(h,m)}} _{2q} &= \lambda_{2q} e^{-\lambda^2_{2q}t} \sum_{k \in \Lambda} a_{(k,2q)} \chi_{2q-1}\eta_{2q-1} \phi_{k,2q} e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}, \tag{24}\\[4pt] {w}^{\text{(h,r)}} _{2q} &= -\lambda^{-1}_{2q} e^{-\lambda^2_{2q}t} \sum_{k \in \Lambda} \Delta\bigl( a_{(k,2q)} \chi_{2q-1}\eta_{2q-1} \phi_{k,2q} \bigr) e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}\notag\\ &\quad - 2\mathrm{i} e^{-\lambda^2_{2q}t} \sum_{k \in \Lambda} \nabla\bigl( a_{(k,2q)} \chi_{2q-1}\eta_{2q-1} \phi_{k,2q} \bigr) \cdot k \, e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}\notag \\ &\quad - \lambda^{-3}_{2q} e^{-\lambda^2_{2q}t} \Delta \Bigl( \sum_{k \in \Lambda} \nabla^{\perp}\bigl( a_{(k,2q)} \chi_{2q-1}\eta_{2q-1} \phi_{k,2q} \bigr) \, \operatorname{curl}\bigl( e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}\bigr) \Bigr).\tag{25} \end{align}\]

• Thanks to \(\Delta f=\text{div}(\nabla+\nabla^{\mathsf{T}})f\) for \(\text{div}f=0\), we infer from 20 that \[\begin{align} \label{def-Rw} {w}^{\text{(h)}} _{2q} &= -\lambda^{-3}_{2q} e^{-\lambda^2_{2q }t} \, \text{div}(\nabla+\nabla^{\mathsf{T}}) \, \nabla^{\perp} \Bigl( \sum_{k \in \Lambda} a_{(k,2q)} \, \eta_{2q-1}\chi_{2q-1} \, \phi_{k,2q} \; \operatorname{curl} \bigl( e^{\mathrm{i}\lambda_{2q } k \cdot x} {k}^{\perp}\bigr) \Bigr) \notag\\ &=:\text{div}(\mathcal{R} {w}^{\text{(h)}} _{2q}), \end{align}\tag{26}\] which shows that \(\eta^2_{2q}\chi^2_{2q}\mathcal{R} {w}^{\text{(h)}} _{2q}=\mathcal{R} {w}^{\text{(h)}} _{2q}\).

We now define the inverse‑cascade dominated flow \({w}^{\text{(i)}} _{2q}\) by \[\label{wttq} {w}^{\text{(i)}} _{2q} = \frac{1}{2} \chi^2_{2q-1}\eta^2_{2q-1} \sum_{k \in \Lambda} \text{div}\Bigl( a^2_k \bigl( \mathrm{Id} + \tfrac{ \mathcal{R} {w}^{\text{(h)}} _{2q-1} }{ 1000C_0} \bigr) (2000C_0) \, {k}^{\perp}\otimes {k}^{\perp}\Bigr) e^{-2\lambda^2_{2q }t}.\tag{27}\] Applying Lemma 2 and the property \(\chi^2_{2q-1}\eta^2_{2q-1} {w}^{\text{(h)}} _{2q-1} = {w}^{\text{(h)}} _{2q-1}\), we obtain \[\begin{align} {w}^{\text{(i)}} _{2q} &= \frac{1}{2} \chi^2_{2q-1}\eta^2_{2q-1} \text{div}\bigl( \mathrm{Id} + \tfrac{\mathcal{R} {w}^{\text{(h)}} _{2q-1}}{1000C_0} \bigr) (2000C_0) e^{-2\lambda^2_{2q }t} \notag \\ &= \chi^2_{2q-1}\eta^2_{2q-1} \, {w}^{\text{(h)}} _{2q-1} \, e^{-2\lambda^2_{2q }t} = {w}^{\text{(h)}} _{2q-1} \, e^{-2\lambda^2_{2q }t}. \label{def-ws} \end{align}\tag{28}\] From this identity, we immediately deduce \[\begin{align} \label{wss2} {w}^{\text{(h)}} _{2q-1}(0,x) = {w}^{\text{(i)}} _{2q}(0,x), \qquad \partial_t {w}^{\text{(i)}} _{2q} = -2\lambda^2_{2q } e^{-2\lambda^2_{2q }t} {w}^{\text{(h)}} _{2q-1} + \partial_t {w}^{\text{(h)}} _{2q-1} \, e^{-2\lambda^2_{2q }t}. \end{align}\tag{29}\]

4.2 Main estimates for \({w}^{\text{(h)}} _{2q}\) and \({w}^{\text{(i)}} _{2q}\)↩︎

In this section we provide preliminary estimates for \({w}^{\text{(h)}} _{2q}\) and \({w}^{\text{(i)}} _{2q}\).

4.3 Construction of \({w}^{\text{(m)}} _{2q}\) and \({d}^{\text{(m)}} _{2q}\)↩︎

Suppose that \((u_{2q-2}, B_{2q-2})\) is a global solution to the MHD equations 1 . We expect that after adding the increments, the resulting pair remains a solution; specifically, let \[u_{2q} = u_{2q-2} + {w}^{\text{(h)}} _{2q} + {w}^{\text{(i)}} _{2q} + {w}^{\text{(m)}} _{2q}, \qquad B_{2q} = B_{2q-2} + {d}^{\text{(m)}} _{2q}.\] Then, in order for \((u_{2q}, B_{2q})\) to satisfy the MHD equations, using Proposition 7, we impose that \(( {w}^{\text{(m)}} _{2q}, {d}^{\text{(m)}} _{2q})\) satisfy the following equations. \[\label{e:wt}\left\{ \begin{alignedat}{-1} &\partial_t {w}^{\text{(m)}} _{2q}-\Delta {w}^{\text{(m)}} _{2q}+ (u_{2q-2} + {w}^{\text{(h)}} _{2q}+ {w}^{\text{(i)}} _{2q}+ {w}^{\text{(m)}} _{2q} )\cdot\nabla {w}^{\text{(m)}} _{2q} + {w}^{\text{(m)}} _{2q}\cdot\nabla(u_{2q-2}+ {w}^{\text{(h)}} _{2q}+ {w}^{\text{(i)}} _{2q}) \\ &\qquad\qquad\qquad\quad= -\nabla p^{(\mathrm{m})}_{2q } + {d}^{\text{(m)}} _{2q}\cdot\nabla {d}^{\text{(m)}} _{2q}+B_{2q-2}\cdot\nabla {d}^{\text{(m)}} _{2q} + {d}^{\text{(m)}} _{2q}\cdot\nabla B_{2q-2} -E_{2q }-F_{2q}, \\ &\partial_t {d}^{\text{(m)}} _{2q}-\Delta {d}^{\text{(m)}} _{2q}+ (u_{2q-2}+ {w}^{\text{(h)}} _{2q}+ {w}^{\text{(i)}} _{2q}+ {w}^{\text{(m)}} _{2q})\cdot\nabla {d}^{\text{(m)}} _{2q} + {w}^{\text{(m)}} _{2q}\cdot\nabla B_{2q-2} \\ &\qquad\qquad\qquad\qquad\qquad= {d}^{\text{(m)}} _{2q}\cdot\nabla (u_{2q-2}+ {w}^{\text{(h)}} _{2q}+ {w}^{\text{(i)}} _{2q}+ {w}^{\text{(m)}} _{2q}) +B_{2q-2}\cdot\nabla {w}^{\text{(m)}} _{2q}-G_{2q} ,\\ & \text{div} {w}^{\text{(m)}} _{2q}=\text{div} {d}^{\text{(m)}} _{2q}= 0, \\ & ( {w}^{\text{(m)}} _{2q}, {d}^{\text{(m)}} _{2q}) |_{t=0}= (0,\lambda^{-50}_{2q-1}f(x) ) , \end{alignedat}\right.\tag{38}\] where \(E_{2q}\) is given by ?? , \(F_{2q}\) is defined by \[\begin{align} F_{2q}=&(\partial_t {w}^{\text{(h)}} _{2q} -\Delta {w}^{\text{(h)}} _{2q})-\Delta {w}^{\text{(i)}} _{2q} \notag \\ &+\text{div}\Big( {w}^{\text{(h)}} _{2q} \otimes( {w}^{\text{(i)}} _{2q}+u_{2q-2}) + ( {w}^{\text{(i)}} _{2q}+u_{2q-2})\otimes {w}^{\text{(h)}} _{2q} \notag \\ &+ {w}^{\text{(i)}} _{2q} \otimes u_{2q-2}+u_{2q-2} \otimes {w}^{\text{(i)}} _{2q}+ {w}^{\text{(i)}} _{2q} \otimes {w}^{\text{(i)}} _{2q}\Big).\label{def-F} \end{align}\tag{39}\] and \[\begin{align} \label{def-G} G_{2q}:=( {w}^{\text{(h)}} _{2q}+ {w}^{\text{(i)}} _{2q})\cdot\nabla B_{2q-2}-B_{2q-2}\cdot\nabla( {w}^{\text{(h)}} _{2q}+ {w}^{\text{(i)}} _{2q}). \end{align}\tag{40}\] In fact, for such \({w}^{\text{(h)}} _{2q}, {w}^{\text{(i)}} _{2q}\) and the given initial conditions, the forced MHD equations 38 are globally well-posed. To establish this result, we first provide some estimates for the forcing terms \(E_{2q}\), \(F_{2q}\), and \(G_{2q}\).

4.4 Estimates of \({w}^{\text{(m)}} _{2q}\) and \({d}^{\text{(m)}} _{2q}\)↩︎

Based on the above estimates for the forcing terms \(E_{2q}, F_{2q}, G_{2q}\), we now formulate the global-in-time existence of solutions to the system 38 as follows.

4.5 Verifying the \(2q-\)th iteration↩︎

Now let us finish the proof of Proposition 4. With the aid of 20 , Propositions 5, 6 and 11, one immediately deduces that \({w}^{\text{(h)}} _{2q}\), \({w}^{\text{(i)}} _{2q}\), \({w}^{\text{(m)}} _{2q}\) and \({d}^{\text{(m)}} _{2q}\) satisfy 12 –?? at \(2q\)-level. Let \[\begin{align} w_{2q}= {w}^{\text{(h)}} _{2q}+ {w}^{\text{(i)}} _{2q}+ {w}^{\text{(m)}} _{2q}, \end{align}\] and \[\begin{align} u_{2q}=u_{2q-2}+w_{2q},\quad B_{2q}=B_{2q-2}+ {d}^{\text{(m)}} _{2q}. \end{align}\] We immediately show 10 and 11 for \(j=2q\). Thanks to Propositions 5 and 11, we have \[\begin{align} &\|(w_{2q}, {d}^{\text{(m)}} _{2q})\|_{L^2_TL^2}\le \lambda^{-15}_{2q-1}\le 2C^{3/4}_02^{-\frac{2q-1}{2}},\\ & \|(w_{2q}, {d}^{\text{(m)}} _{2q})\|_{L^\infty_t \dot{B}^{0}_{\infty,1}\cap L^1_t\dot{B}^2_{\infty,1}}\le 2 C^{3/4}_0 \lambda_{2q}+\lambda^{2r}_{2q}\le \frac{1}{2}C_0\lambda_{2q},\\ &\|(w_{2q}, {d}^{\text{(m)}} _{2q})\|_{L^1_t \dot{B}^{1}_{\infty,1}}\le 2C^{3/4}_0+ \lambda ^{-20}_{2q-1}\le C_0. \end{align}\] The above two estimates combined with 8 –9 for \(j=2q-1\) show that 8 and 9 hold for \(j=2q\). Hence, we complete the proof of Proposition 4 for the case where \(m\) is even. For the case that \(m\) is odd, we have \(m=2q+1\) for some \(q \ge 1\). We construct \({w}^{\text{(h)}} _{2q+1}, {w}^{\text{(i)}} _{2q+1}\), \({w}^{\text{(m)}} _{2q+1}\) and \({d}^{\text{(m)}} _{2q+1}\) by replacing the parameter \(2q\) in definitions 20 , 28 and in the system 38 with \(2q+1\). Let \(w_{2q+1}= {w}^{\text{(h)}} _{2q+1}+ {w}^{\text{(i)}} _{2q+1}+ {w}^{\text{(m)}} _{2q+1}\), \(u_{2q+1}=u_{2q-1}+w_{2q+1}\) and \(B_{2q+1}=B_{2q-1}+ {d}^{\text{(m)}} _{2q+1}\). Following the same argument as for the even case, Proposition 4 is established also for odd \(m\), which finishes the proof of Proposition 4.

Acknowledgement↩︎

We thank Professor Mimi Dai for the valuable comments on this manuscript. This work was supported by the National Key Research and Development Program of China (Grant number No. 2022YFA1005700) and the National Natural Science Foundation of China (Grant numbers No.12371095, No. 12401277 and No. 12501304).

References↩︎